On the convergence of multistep methods for nonlinear stiff differential equations

Summary Convergence estimates are given forA()-stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems. The approach taken here combines perturbation arguments with frequency domain techniques.     

Note onA-stability of multistep multiderivative methods

Daniel and Moore [4] conjectured that anA-stable multistep method using higher derivatives cannot have an error order exceeding 2l. We confirm partly this conjecture by showing that for a large class ofA-stable methods the error order can not be 2l+1 mod 4. This extends results found in Jeltsch [13].     

On the Convergence Behaviour of Variable Stepsize Multistep Methods for Singularly Perturbed Problems

In this note, we investigate the convergence behaviour of linear multistep discretizations for singularly perturbed systems, emphasising the features of variable stepsizes. We derive a convergence result for A()-stable linear multistep methods and specify a refined error estimate for backward differentiation formulas. Important ingredients in our convergence analysis are stability bounds for non-autonomous linear problems that are obtained by perturbation techniques.     

A new class of highly-stable methods:A 0-stable methods

A linear multistep method (,) is defined to beA ₀-stable if when it is applied to the equation the approximate solutionx ^h (t _n ) tends to zero ast _n for all values of the stepsizeh and all(0, ).Various properties ofA ₀-stable methods are derived. It is shown that most of the properties ofA()-stable methods are shared byA ₀-stable methods. It is proved that there existA ₀-stable methods of arbitrarily high order.Sponsored by the Office of Naval Research under Contract No.: N00014-67-A-0128-0004, and the United Kingdom Science Research Council.     

A- andB-stability for Runge-Kutta methods-characterizations and equivalence

Summary Using a special representation of Runge-Kutta methods (W-transformation), simple characterizations ofA-stability andB-stability have been obtained in [9, 8, 7]. In this article we will make this representation and their conclusions more transparent by considering the exact Runge-Kutta method. Finally we demonstrate by a numerical example that for difficult problemsB-stable methods are superior to methods which are onlyA-stable.Talk, presented at the conference on the occasion of the 25th anniversary of the founding ofNumerische Mathematik, TU Munich, March 19–21, 1984     

Norm bounds for rational matrix functions

Summary If the field of values of a matrixA is contained in the left complex halfplaneH and a functionf mapsH into the unit disc then f(A)₂1 by a theorem of J.v. Neumann. We prove a theorem of this type, only the field of values ofA is used for functions which are absolutely bounded by one in only part ofH. An extension can be used to show norm-stability of single step methods for stiff differential equations. The results are applicable among others to several subdiagonal Padé approximations which are notA-stable.     

Méthodes multipas pour des équations paraboliques non linéaires

Summary We study the error due to the discretization in time of a nonlinear parabolic problem by a multistep method. Error estimates are obtained if the method is of the orderp (p>1) and stronglyA()-stable . The method is also applied to the Navier-Stokes equations in two dimensions.

     

A 0-Stability and stiff stability of Brown's multistep multiderivative methods

Summary Brown [1] introducedk-step methods usingl derivatives. Necessary and sufficient conditions forA ₀-stability and stiff stability of these methods are given. These conditions are used to investigate for whichk andl the methods areA ₀-stable. It is seen that for allk andl withk1.5 (l+1) the methods areA ₀-stable and stiffly stable. This result is conservative and can be improved forl sufficiently large. For smallk andl A ₀-stability has been determined numerically by implementing the necessary and sufficient condition.     

Stability properties of collocation methods

Lower bounds for are given for which equidistant s-point collocation methods areA()-stable for arbitrarys.     

A criterion forP-stability properties of Runge-Kutta methods

P-stability is an analogous stability property toA-stability with respect to delay differential equations. It is defined by using a scalar test equation similar to the usual test equation ofA-stability. EveryP-stable method isA-stable, but anA-stable method is not necessarilyP-stable. We considerP-stability of Runge-Kutta (RK) methods and its variation which was originally introduced for multistep methods by Bickart, and derive a sufficient condition for an RK method to have the stability properties on the basis of an algebraic characterization ofA-stable RK methods recently obtained by Schere and Müller. By making use of the condition we clarify stability properties of some SIRK and SDIRK methods, which are easier to implement than fully implicit methods, applied to delay differential equations.     

On the stability of Runge-Kutta methods for delay integral equations

Summary We present a class of Runge-Kutta methods for the numerical solution of a class of delay integral equations (DIEs) described by two different kernels and with a fixed delay . The stability properties of these methods are investigated with respect to a test equation with linear kernels depending on complex parameters. The results are then applied to collocation methods. In particular we obtain that any collocation method for DIEs, resulting from anA-stable collocation method for ODEs, with a stepsize which is submultiple of the delay , preserves the asymptotic stability properties of the analytic solutions.This work was supported by CNR (Italian National Council of Research)     

Smoothing Properties and Approximation of Time Derivatives for Parabolic Equations: Variable Time Steps

We study smoothing properties and approximation of time derivatives for time discretization schemes with variable time steps for a homogeneous parabolic problem formulated as an abstract initial value problem in a Banach space. The time stepping methods are based on using rational approximations to the exponential function which are A()-stable for suitable (0,/2] with unit bounded maximum norm. First- and second-order approximations of time derivatives based on using difference quotients are considered. Smoothing properties are derived and error estimates are established under the so-called increasing quasi-quasiuniform assumption on the time steps.     

Error bounds for continued fractionsK(1/b n)

Summary A priori truncation error bounds are obtained for continued fractions of the formK(1/b _n),b _n complex. The error bounds are easily applied to the case whenb _n0 asn. A numerical example involving the complex error function is given.     

On the convergence of the symmetric SOR method for matrices with red-black ordering

Summary We prove that if the matrixA has the structure which results from the so-called red-black ordering and ifA is anH-matrix then the symmetric SOR method (called the SSOR method) is convergent for 0<<2. In the special case thatA is even anM-matrix we show that the symmetric single-step method cannot be accelerated by the SSOR method. Symmetry of the matrixA is not assumed.     

Stability of Numerical Methods for Ordinary Differential Equations

Variable stepsize stability results are found for three representative multivalue methods. For the second order BDF method, a best possible result is found for a maximum stepsize ratio that will still guarantee A(0)-stability behaviour. It is found that under this same restriction, A()-stability holds for 70°. For a new two stage two value first order method, which is L-stable for constant stepsize, A(0)-stability is maintained for stepsize ratios as high as aproximately 2.94. For the third order BDF method, a best possible result of (1/2)(1+ ) is found for a ratio bound that will still guarantee zero-stability.     

Some new results on Unsymmetric Successive Overrelaxation method

Summary The Unsymmetric Successive Overrelaxation (USSOR) iterative method is applied to the solution of the system of linear equationsA x=b, whereA is annxn nonsingular matrix. We find the values of the relaxation parameters and for which the USSOR iterative method converges. Then we characterize those matrices which are equimodular toA and for which the USSOR iterative method converges.     

Strong Stability and Non-smooth Data Error Estimates for Discretizations of Linear Parabolic Problems

We study smoothing properties of discretizations of a linear parabolic initial boundary value problem with a possibly non-selfadjoint elliptic operator. The solution at time t > 0 of this problem, as well as its time derivatives, are in L _r for initial values in L _s even when r > s. We show that similar strong stability results hold for discrete solutions obtained by discretizing in space by linear finite elements and in time by a class of A()-stable implicit rational multistep methods (including single step methods as a special case) with good smoothing properties, as well as for certain combinations of single step methods. Most of our results are derived from the corresponding L ₂-bounds, shown by semigroup techniques, together with a discrete Gagliardo-Nirenberg inequality, and generalize previously known estimates with respect to admissible problems and time discretization methods. Our techniques make it possible to obtain, e.g., supremum norm error estimates for initial data which are only required to be in L ₁.     

Numerical treatment of O.D.Es.: The theory ofA-methods

Summary Almost all commonly used methods for O.D.Es. and their most miscellaneous compositions areA-methods, i.e. they can be reduced toz _o=;z _j =Az _j–1 +h(x _j–1 ,z _j–1 ,z _j ;h),z _j ^s ,A(s,s),j=1(1)m. This paper presents a general theory forA-methods and discusses its practical consequences. An analysis of local discretization error (l.d.e.) accumulation results in a general order criterium and reveals which part of the l.d.e. effectively influences the global error. This facilitates the comparison of methods and generalizes considerably the concept of error constants. It is shown, as a consequence, that the global error cannot be safely controlled by the size of the l.d.e. and that the conventional error control may fail in important cases. Furthermore, Butcher's effective order methods, the concept of Nordsieck forms, and Gear's interpretation of lineark-step schemes as relaxation methods are generalized. The stability of step changing is shortly discussed.     

A perturbation bound for the generalized polar decomposition

LetA be anm×n complex matrix. A decompositionA=QH is termed ageneralized polar decomposition ofA ifQ is anm×n subunitary matrix (sometimes also called a partial isometry) andH a positive semidefinite Hermitian matrix. It was proved that a nonzero matrixA ^m×n has a unique generalized polar decompositionA=QH with the property (Q ^H )=(H), whereQ ^H denotes the conjugate transpose ofQ and (H) the column space ofH. The main result of this note is a perturbation bound forQ whenA is perturbed.     

On quadratic and sesquilinear functionals

LetX be a leftA-module, whereA is either a complex Banach *-algebra with an identity element or the field of quaternions. The main result of this note is that forQ, anA-quadratic functional defined onX, there exists a sesquilinear functionalB such thatB(x,x)=Q(x) holds for allxX.